On The Performance of the Logistic-Growth Population Projection Models

Transcription

1 On The Performance of the Logistic-Growth Population Projection Models Eguasa O., Obahiagbon K. O. and Odion A. E. Department of Mathematics and Computer Science, Benson Idahosa University, Benin City. Nigeria. Abstract In this paper, we proposed two mathematical Models for population projection, the exponential growth and the Logistic Model. Thereafter, both Models were transformed and since Logistic Model is not linear in its parameters hence a surgery was performed on the data values separated by a fixed time say This enables the Logistic parameters to be sometimes estimated by Least-Squares. It was shown that the estimates were quite close to the data set from the National Population Commission of Nigeria. Keywords: Carrying capacity, Logistic Model, Exponential growth, Polyfit, Least- Squares, Population Projection. 1.1 Introduction One important area where data collection for human population is so vital is basically for national planning or budgeting. This is while government of nations placed emphasis on Census to determine the population count. Though, this exercise is quite expensive, time consuming and yet is still one of the major ground for population count. This explains the seriousness attached to Census by government for developmental purposes. Population data gives an insight about the Social- economic conditions of a country; hence the need for projection is usually based on mathematical models which are significant. These models use vital assumptions about its parameters values that reflect the true nature of the population for which these assumptions were made. Choi (2010) proposed an application of the component model to development of local population projections derived from the housing unit method. Haque et al. (2012) analyzed fourth order Rung-Kutta Scheme for the numerical of the non-autonomous and non-linear model to incorporate the growth rate as a function of time. Wali et al. (2012) focuses on the application of logistic equation to model the population growth of Uganda using data from 1980 to Earterling et al. (2000) introduced the integral projection model which eliminates the need for division into discrete classes, without requiring any additional biological assumptions. National Population Commission of Nigeria (2005), National Policy on Population for Sustainable Development. Neverova and Frisman (2012) investigate a two-component model of population dynamics with seasonal reproduction. Akamine and Suda (2011) analyzed the population projection matrix models in random walk models and the growth rate of the mean population size, which is equal to the maximum eigenvalue of the mean matrix, is better than the average of the intrinsic rates of natural increase calculated by computer simulation. Simpson reviews the mechanisms which can incorporate into a cohort component population projection incomplete and estimated data since the base year, as well as targets for population, housing and employment. 2.1 The Growth Model Consider the differential equation,.(1)...(2).(3)...(4) is the initial population..(5) 102

2 From equation (5)..(5a)...(5b) Equation (5) gives the solution to the differential equation (1) and equation (5b) is Linear in its parameters. Equation (5) gives the population size accurately after some time t, thereafter the population goes unbounded resulting to explosion in the population. Hence, we now consider the Logistic Model. 2.2 Logistic Model Consider the model by Verhulst in 1845, Yeargers et al. (1996),..(6) Where, y = population size K = demographic indicator (Carrying Capacity) r = growth Rate. It is easy to see that, when y = 0 or y = k. These are the stationary points of the equation (6). The stationary point y = k, at which the per capita growth rate becomes zero is called the carrying capacity of the environment. 2.3 Solution to the Model.(6).(7) Decomposing the L.H.S of equation (7) by partial fraction ( )...(8)... ( 9) Reciprocating both sides, Isolating y, (10)...(11) Equation (12) is the solution to the differential equation (6).But, equation (6) with initial population can be obtained..(12).(13) Logistic parameters can sometimes be estimated by Least Squares. Equation (12) is not Linear in its parameters A, r and K. If the data values are separated by fixed time, then Least Squares can be applicable. Suppose the data points are and the predicted value of from equation (10) 103

3 Mathematical Theory and Modeling...(14) ( ) (15).. (16)....(17) Substitute equation (17) into equation (15) ( )..(18)...(19) Set in equation (20)....(20)....(21).....(22) Therefore, a Least-Square is performed on the points, to determine r and k. With r and k known, A can be determined from equation (10) to obtain an estimate of z. From equation (22), set, (23) We shall perform Least-Squares on equation (23) with the set of data using MATLAB inbuilt function. 3.1 Analysis of Model for Data I CASE 1: Consider the solution from equation (5b) and applying MATLAB inbuilt function on the set of data for the population projection system of Male in KANO STATE, NIGERIA. Applying MATLAB inbuilt function to estimate equation (5b) >> t=[ ]; >> z=[ ]; >> logy=(z); >> p=polyfit(t,logy,1) p = 1.0e+006 * (24) CASE 2: Consider the solution from equation (23)....(23) applying Least-Squares on and using MATLAB inbuilt function on the set of data for the population projection system of Male in KANO STATE, NIGERIA.As in Table

4 Mathematical Theory and Modeling >> x=[ ]; >> y=[ ]; >> p=polyfit(x,y,1) p = >> xp= :10000: ; >> zp=polyval(p,xp) >> plot(x,z,'o',xp,zp) From equation (23).(25) Recall, with r and k known A can be determined from equation (10). CASE 3: Consider equation (10).(10) From equation (10), we have...(26) From equation (24), (25) and (26) we have,...(24)..(25)...(26) A projection can be obtained from the three equations above to estimate the given data z. 3.2 Analysis of Model for Data II CASE 1: Consider the solution from equation (5b) and applying MATLAB inbuilt function on the set of data for the population projection system for Male and Female in KANO STATE, NIGERIA. Applying MATLAB inbuilt function to estimate equation (5b) >> t=[ ]; >> z=[ ]; >> logy=(z); >> p=polyfit(t,logy,1) p = 1.0e+006 * (27) CASE 2: Consider the solution from equation (23)...(23) 105

5 Mathematical Theory and Modeling applying Least-Squares on and using MATLAB inbuilt function on the set of data for the population projection system for both Male and Female in KANO STATE, NIGERIA.As in Table 3. >> x=[ ]; >> y=[ ]; >> p=polyfit(x,y,1) p = >> xp= :10000: ; >> yp=polyval(p,xp) >> plot(x,y,'o',xp,yp) From equation (25),.(28) Recall, CASE 3: Consider equation (10) with r and k known A can be determined from equation (10). (10) From equation (10), we have..(29) Therefore, (27) (28)...(29) 3.3 Analysis of Model for Data III CASE 1: Consider the solution from equation (5b) and applying MATLAB inbuilt function on the set of data for the population projection system for both Male and Female in AKWA IBOM STATE, NIGERIA. Applying MATLAB inbuilt function to estimate equation (5b) >> t=[ ]; >> z=[ ]; >> logy=(z); >> p=polyfit(t,logy,1) p = 1.0e+006 * (30) 106

6 Mathematical Theory and Modeling CASE 2: Consider the solution from equation (23).(23) applying Least-Squares on and using MATLAB inbuilt function on the set of data for the population projection system for both Male and Female in AKWA-IBOM STATE, NIGERIA.As in Table 5. >> x=[ ]; >> y=[ ]; >> p=polyfit(x,y,1) p = >> xp= :10000: ; >> yp=polyval(p,xp) >> plot(x,y,'o',xp,yp).(31) Recall, CASE 3: Consider equation (10) with r and k known A can be determined from equation (10)...(10) and from equation (10) we have,.(32) Therefore,...(30)...(31)...(32) 4. Conclusion In this paper a mathematical estimation for the population projection of three states in Nigeria was analyzed based on two ordinary differential equations model which are the growth model and the logistic model. Firstly, the growth model was transformed to linear model whose parameters were estimated. Secondly, it was established that the solution to the logistic model is not linear in its parameters A, r and k. Therefore a surgery was performed on the data values which are separated by fixed time, and then Least Squares can be applicable. Thereafter, three equations were obtained to estimate the data from the National Population commission of Nigeria. The data values from the National population Commission of Nigeria compares well with the estimated population model, and in each of the states considered. We also consider the MATLB inbuilt function to polyfit the data set, and obtained a linear plot for the data set. It was discovered that has the best approximation to the population data. 107

7 Appendix Table 1: Population Data YEAR (t) MALE IN KANO STATE(DATA I) MALE & FEMALE IN KANO STATE(DATA II) Source: National Population Commission of Nigeria, MALE & FEMALE IN AKWA- IBOM STATE (DATA III) Table 2: Population of Male in KANO STATE, NIGERIA. YEAR (t) POPULATION (z)

8 Table 3 : Population projection of Male in KANO STATE, NIGERIA. YEAR (t) POPULATION (z) Table 4: Population for both Male and Female in KANO STATE, NIGERIA. YEAR (t) POPULATION (z) Table 5: Population projection for both Male and Female in KANO STATE, NIGERIA. YEAR (t) POPULATION (z)

9 Table 6: Population for both Male and Female in AKWA-IBOM STATE, NIGERIA. YEAR (t) POPULATION (z) Table 7: Population projection for both Male and Female in AKWA-IBOM STATE, NIGERIA. YEAR (t) POPULATION (z)

10 Figure 1: Linear plot for in data I. Figure 2: Linear plot for in data II. 111

11 6 x x 10 6 Figure 3: Linear plot for in data III. References Akamine, T. and Suda, M. (2011):The Growth Rates of Population Projection Matrix Models in Random Environments.Aqua-BioScience Monographs, Vol. 4(3), pp Brewer, R. (1988): Population Ecology: The Science of Ecology, Saunders College Publishing, Ft. Worth, 2 nd ed. Choi, S. (2010): Application of the Cohort Component Model to Development of local population projections. Association of Collegiate of Planning Conference. Eartering, R. M., Ellner, P. S. and Dixon, M. P. (2000): Size-Specific Sensitivity: Applying a Structured Population Model. Ecology, 81(3), pp Haque, M. M., Ahmed, F., Anam, S. and Kabir, R. M. (2012): Future Population Projection of Bangladesh by Growth Rate Modeling Using Logistic Population Model, Vol. 1(2), pp National Population Commission of Nigeria.National Policy on Population for Sustainable Development. (2005). Noverova, G. and Frisman, E. (2012): Mathematical Model for Number Dynamics of Population with Varying Reproduction Age. International Environmental Modeling and Software Society (iemss). Simpson, L.Integrated Estimates and Targets within a Population Projection. ESRC Research Methods Programme Working Paper No. 18. Wali, A., Kagoyire, K. and Icyingeneye, P. (2012): Mathematical Modeling of Uganda Population Growth. Applied Mathematical Sciences, Vol.6( 84), pp Yeargers, K. E., Shonkwiler, W. R. and Herod, V. J. (1996): An Introduction to the Mathematics of Biology with Computer Algebra Models. 112

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